The ${B}^{+}\ensuremath{\rightarrow}{D}^{+}{K}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ decay is observed in a data sample corresponding to $3.0\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ of $pp$ collision data recorded by the LHCb experiment during 2011 and 2012. The signal significance is $8\ensuremath{\sigma}$ and the branching fraction is measured to be $\mathcal{B}({B}^{+}\ensuremath{\rightarrow}{D}^{+}{K}^{+}{\ensuremath{\pi}}^{\ensuremath{-}})=(5.31\ifmmode\pm\else\textpm\fi{}0.90\ifmmode\pm\else\textpm\fi{}0.48\ifmmode\pm\else\textpm\fi{}0.35)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$, where the uncertainties are statistical, systematic and due to the normalization mode ${B}^{+}\ensuremath{\rightarrow}{D}^{\ensuremath{-}}{K}^{+}{\ensuremath{\pi}}^{+}$, respectively. The Dalitz plot appears to be dominated by broad structures. Angular distributions are exploited to search for quasi-two-body contributions from ${B}^{+}\ensuremath{\rightarrow}{D}_{2}^{*}(2460{)}^{0}{K}^{+}$ and ${B}^{+}\ensuremath{\rightarrow}{D}^{+}{K}^{*}(892{)}^{0}$ decays. No significant signals are observed and upper limits are set on their branching fractions.